An intercomparison of models predicting growth of Antarctic krill (Euphausia superba): The importance of recognizing model specificity

Antarctic krill (Euphausia superba) is a key species of the Southern Ocean, impacted by climate change and human exploitation. Understanding how these changes affect the distribution and abundance of krill is crucial for generating projections of change for Southern Ocean ecosystems. Krill growth is an important indicator of habitat suitability and a series of models have been developed and used to examine krill growth potential at different spatial and temporal scales. The available models have been developed using a range of empirical and mechanistic approaches, providing alternative perspectives and comparative analyses of the key processes influencing krill growth. Here we undertake an intercomparison of a suite of the available models to understand their sensitivities to major driving variables. This illustrates that the results are strongly determined by the model structure and technical characteristics, and the data on which they were developed and validated. Our results emphasize the importance of assessing the constraints and requirements of individual krill growth models to ensure their appropriate application. The study also demonstrates the value of the development of alternative modelling approaches to identify key processes affecting the dynamics of krill. Of critical importance for modelling the growth of krill is appropriately assessing and accounting for differences in estimates of food availability resulting from alternative methods of observation. We suggest that an intercomparison approach is particularly valuable in the development and application of models for the assessment of krill growth potential at circumpolar scales and for future projections. As another result of the intercomparison, the implementations of the models used in this study are now publicly available for future use and analyses.

In this part, we validate our krill growth model implementations for correctness. This is done by running the different models using the environmental input data used in their original publication. We then compare the predicted growth trajectories with those that were originally reported. Original data (environmental data and growth trajectories) were extracted from the original manuscripts using a data extraction tool from ImageJ.
Hofmann and Lascara (2000) In the quality control of our implementation, we simulated the growth trajectories of 3 individuals with starting lengths of 2, 22 and 45mm over 468 days as was done in the original paper. The simulation was performed twice, once excluding ice algae as a food source (trajectories shown in Figure 7 of the paper) and including ice algae ( Figure 8 of the paper). The model simulations slightly diverge from the original trajectories but general patterns are reasonably well reproduced -possible explanations for the divergences could come from differences in the numerical integration, inaccuracies of our environmental time series which was extracted from Figure 6 or minor differences in parameter values that are reported in the original manuscript and the Fortran-files of the model that Eileen Hofmann provided us with. The Fortran-files include minor modifications of the model that were not described in the manuscript. We did not include these modifications as they actually worsen the fit (modifications are that daily rations are capped at 20% and set to 0 below 1.5%) Including ice algae as a second food source results in stronger differences between our implementation and the originally reported trajectories. Growth is strongly overestimated in our version. We went through the Fortran files and could not really find the problem yet. Unfortunately, we do not have access to the original model input files which are being used in the Fortran-script to parameterize ingestion and assimilation of sea ice biota. We also could not find any mistakes in our implementation.
In our project, we restrict the simulations to summer growth and excluded ice algae as a food source -also because no circumpolar predictions of sea ice biota (accessible for krill) exist. Therefore, potential inaccuracies in the circumpolar model simulations are comparable with those shown in the left panel of Figure 1.
Conclusion: We conclude that our implementation reasonably well reproduces growth trajectories that the original model would predict. For future studies, it would be good to identify the differences between our model version and the original model, especially regarding the inclusion of sea ice biota as a food source.

Ryabov et al. (2017)
The growth component of the Ryabov et al. (2017) model was directly provided by Alexey Ryabov. He shared the growth component of his model using an environmental data time series that we previously provided him with. His model version was implemented in Matlab files that we translated into R.   2002) has a temperature-dependence of metabolism added, otherwise both models are identical), we assume that the quality assessment yields similar results as the one on Hofmann and Lascara (2000).

Atkinson et al. (2006):
In Atkinson et al. (2006), no growth trajectories are shown but predicted daily growth rates for individuals of three different length at two temperatures over a chlorophyll a gradient are presented. We compare the predictions of our implementations (black lines) with those shown in the Figure 6 in the manuscript (red points).
The predicted growth rates from our implementation perfectly match those reported in the paper.
Conclusion: The model is correctly implemented and can be used as it is.

Tarling et al. (2006)
The original manuscript does not provide growth trajectories. However, it provides model estimates for intermoult periods for different stages, environmental conditions and body lengths in Figure 7 and 8 of the original manuscript.   2006)) was not visualized in its original publication, but given the relative simplicity of this function, it is unlikely that we have implemented it incorrectly.
Add-on: We note that the model of Tarling et al. (2006) produces growth trajectories that differ from those predicted by the model of Atkinson et al. (2006). In the following, we take a more detailed look at the mechanisms causing these differences.
It can be seen that the growth increment function of the model of Atkinson et al. (2006) for juvenile krill has a similar shape than the daily growth rate-function for different temperatures and chlorophyll a concentrations ( Figure 5 2006), the predicted daily growth rates also peak at the lowest temperatures. For completeness, we also compared the growth properties predicted by both models for adult krill.
For adult female krill, we see that the intermoult period function of the model of Tarling et al. (2006) has a different shape to that for juvenile krill. Minimum intermoult periods, and therefore most frequent moultings, are predicted for temperatures around 1°C ( Figure 6). Generally, these intermoult periods are longer than the ones predicted for juvenile krill which results in reduced growth. In addition, the growth increment function from Atkinson et al. (2006) predicts comparatively less growth (¡10% growth increment for all environmental conditions investigated here). Since both the intermoult period function and the growth increment functions take on values corresponding to high growth for temperatures of ∼1°C, the daily growth rates predicted for adult females by the Tarling et al. (2006) model peak around the same temperatures. This is

Wiedenmann et al. (2008)
The model structure of Wiedenmann et al. (2008) is very similar to Tarling et al. (2006) but instead of using the intermoult period model from Tarling et al. (2006), it uses the intermoult period-model presented in Kawaguchi (2006), in which intermoult period is a function of water temperature only. Other than that, both models are identically implemented using the Atkinson et al. (2006) all krill-growth increment function for predicting growth at the time of moulting. We can compare the intermoult period model used in our implementation of Wiedenmann et al. (2008) with the results shown in Figure 5 in Kawaguchi et al. (2006).

Bahlburg et al. (2021)
This model was originally developed and implemented by the corresponding author and is publicly documented under https://github.com/dbahlburg/SERBIK. A minor change from the original model was made in our simulations by removing the "foodConversion"-constant of 0.8 which is multiplied with the ingested carbon. This was done since we realized that this constant is not included in the baseline model of Jager and Ravagnan (2015).

Jager and Ravagnan (2015):
This model assumed unlimited food in its original publication. When using the same assumption and the exact parameterization provided in Table 2 of the manuscript, our implementation systematically underestimates the originally reported daily growth rates as a function of body length at 0°C and 5°C: Increasing the area-specific carbon assimilation constant slightly from 0.044 to 0.045 gives a better match. For our analyses, we worked with this ajusted parameter value. reproduced results (black lines) at 0°C and 5°C using a area-specific carbon assimilation constant of 0.044, right panel: same simulations but using a area-specific carbon assimilation constant of 0.045

Constable and Kawaguchi (2018) -excluded from the main analysis
The model of Constable and Kawaguchi integrates parts of Hofmann and Lascara (2000) into an intermoult period framework. We compare our implementation with the results shown in Figure 4 and 5 in the original paper. It can be seen that in our implementation, growth rates are much higher than in the reported results in the paper. Dry mass is almost 3 times higher, reproduction occurs much more frequently and length increases at substantially higher rates. mass is peaking at the threshold value in our simulation while it remains much lower in the original results. Similarly, the krill is considerably smaller (∼45 mm vs. 65 mm) in the original results. It seems that in our implementation, the krill has much more energy available for growth and reproduction compared to the original simulations. This means that differences must exist in either the mechanisms of food assimilation or respiration. However, the underlying intermoult period-philosophy of the model is merely reflected in the dynamics shown in the original Figure 4 and 5 (we would expect a step-wise growth pattern similar to Tarling et al. 2006or Wiedenmann et al. 2008. Krill growth seems to be continuous, energy dynamics look smooth (although there should be a sudden loss of body mass at the time of moulting due to the discard of the old carapace) and gonad dry mass does not show the dynamics we would expect based on the model mechanisms (gonad dry mass should be reduced to zero after each spawning, however there is only a partial reduction). Apparently, a considerable energy surplus leads to exaggerated growth and reproduction in our model. We took another look at the environmental time series presented in Figure 3 of the paper. According to the authors, it shows the dynamics of POC, temperature and day length used to create the model results shown in Figure 5 and 6 of the original paper. After taking a closer look, we realized that the POC concentrations in this time series are very high. Approximating the concentrations as chlorophyll a concentrations reveals that the krill individual is exposed to concentrations of >5 mg Chla m −3 in at least 6 months (using a conversion constant of ChlA:POC = 50 as in Hofmann and Lascara (2000)). These very high food concentrations could explain some of the differences between our model and the trajectories from the original paper. Arguably, the displayed environmental time series deviates from the actual one used for model simulations. To see how our implementation performs using more realistic food concentration, we reduced the POC concentrations by an arbritary factor of 12 (blue line, the dotted line marks the 1 mg m −3 threshold). This way, approximated chlorophyll a concentrations vary between 0 and 1.4 mg m −3 with an annual average of ∼0.5 mg m −3 . The modified time series might be on the lower end of field chlorophyll a dynamics but still representative of the Southern Ocean environment. We re-ran the simulations with the reduced food concentrations and obtained the following results: In the more detailed analysis, we see that our model typically misses the first two annual spawning events shown in the original results. The number of released eggs in following spawnings are slightly higher in our model (indicated by the higher gonad dry mass). There are strong indications that the originally presented environmental time